The only middle line he can walk in this situation, while trying not to tip one way or the other, is to offer "both sides" some sort of solution:

Dems: protect DREAMERS and try to streamline the process of immigration for asylum-seekers, refugees of political violence/genocide, and applicants without a criminal record.

GOP: Cracking down on illegal crossings and providing assistance to the beaurocratic offices that need the most assistance on the border to process those waiting and turn away those, with dignity and respect, that don't "qualify" for emergency processing.

To help "fix" the "problem": Propose and introduce legislation to fine companies that rely on illegal labor to send a message.

Then publish the names of all major corporations that do it to enforce the message and slap down any claims of not doing anything while still looking out for "America's" best interest.

Anyone who complains can be countered with, "Biden is looking to protect American jobs and 'punish' American companies who don't believe in hiring Americans over illegal immigrants."

Then, it'll become crystal-clear that even if Americans wanted those jobs, they wouldn't do them (and if so, not for the pay undocumented workers are getting).

If you can't fix the problem, it's best to expose it and call out those (GOP) that refuse to work on it but still benefit from it.

Or something like that.